The salts in dog urine will burn both the grass blades and the roots of the plants. The only way to avoid this is to immediately hose off the area with lots of water. I'm afraid that once the lawn is dead in those spots, the grass itself is dead and will need to be replanted. Rake out the affected areas, reseed, and cover with a light layer of peat moss to protect the seeds until they germinate. The good news is that grass seeds germinte in 7-10 days so the fix will be quick. Best wishes with your lawn!